Hegel’s Science of Logic


Remark 1: The Species of Calculation in Arithmetic; Kant's Synthetic Propositions a priori of Intuition

§ 445

Spatial magnitude and numerical magnitude are usually regarded as two species, the former being on its own account a determinate magnitude just as much as the latter; their difference is held to consist only in the different determinations of continuity and discreteness, but as quantum they stand on the same level. In spatial magnitude, geometry has, in general, continuous magnitude for its subject matter while the subject matter of arithmetic is the discrete magnitude of number. But with this dissimilarity of their subject matter, the manner and completeness of their limitation or determinedness is also different. Spatial magnitude possesses only limitation generally; if it is to be considered as a thoroughly determinate quantum then number is required. Geometry as such does not measure spatial figures (it is not mensuration), but only compares them. In its definitions, too, the determinations are in part derived from the equality of the sides and angles, or from equidistance. Thus the circle, because it is based solely on the equidistance of all possible points in it from a centre, does not require number for its determination. These determinations based on equality or inequality are genuinely geometrical. But they are not sufficient, and for other figures, for example, the triangle or rectangle, number is requisite; this in its principle, the one, contains a self-determinedness, it is determined without the aid of an other and therefore not through comparison. It is true that in the point, spatial magnitude has a determinateness corresponding to the one; but the point, in becoming external to itself, becomes an other, becomes the line; because it is essentially only a spatial one, it becomes in the relation a continuity in which the nature of the point, the self-determinedness, the one, is sublated. In so far as the self-determinedness is supposed to be preserved in the self-externality, the line must be represented as an aggregate of ones, and to the limit must be imparted the determination of the many ones, that is, the magnitude of the line — and similarly of other spatial determinations — must be taken as a number.

§ 446

Arithmetic considers number and its figures; or rather does not consider them but operates with them. For number is the determinateness which is indifferent, inert; it must be actuated from without and so brought into a relation. The modes of relation are the species of calculation. In arithmetic they are presented seriatim and it is clear that one depends on the other; but the thread which links the progressive stages is not made prominent in arithmetic. However, the systematic arrangement justly claimed for the presentation of these elements in the textbooks is readily provided by the determinations of number itself stemming from its Notion. These cardinal determinations will be briefly noted here.

§ 447

Number has for its principle the one and is, therefore, simply an aggregate externally put together, a purely analytic figure devoid of any inner connectedness; and because it is produced in this merely external manner all calculation is the production of numbers, a counting or, more specifically, a counting up. Any diversity in this external production, which always proceeds to the same end, can lie only in a difference between the numbers which are to be counted up; this difference must itself come from elsewhere and from an external determination.

§ 448

The qualitative difference which constitutes the determinateness of number is, we have seen, that of unit and amount; consequently, every determinateness of the notion of number which can occur in the species of calculation can be reduced to this difference. But the difference which belongs to numbers as quanta, is external identity and external difference, equality and inequality, which are moments of reflection and fall to be considered under the determinations of essence when we come to deal with difference.

§ 449

Further, we must premise that numbers can, in general, be produced in two ways, either by aggregation, or by separation of an aggregate already given; in both cases the same specific kind of counting is employed, so that to an aggregating of numbers there corresponds what may be called a positive species of calculation, and to a separating of them, a negative species; the determination of the species of calculation itself is independent of this antithesis.

§ 450

1. After these remarks we proceed to indicate the species of calculation. The first production of number is the aggregating of the many as such, each of which then is posited as only a one — numbering or counting. Since the ones are mutually external their representation is illustrated sensuously, and the operation by which number is generated is a process of counting on the fingers, dots, and so on. What four, five, etc., is, can only be pointed out. Seeing that the limit is an external one, the breaking off of the counting, the amount to be aggregated, is contingent, arbitrary. The difference of amount and unit which appears in the progress from one species of calculation to another, establishes a system of numbers, dyadic, decadic, and so on; such a system rests on the whole on the arbitrary choice of an amount which is consistently to be taken as unit.

§ 451

The number produced by counting are in turn themselves counted; and as thus immediately posited they are determined as still lacking any connection with one another, as indifferent to equality or inequality, and their magnitude relatively to one another is contingent — they are, therefore, simply unequal; this is addition. We learn that 7 + 5 = 12 by adding (on our fingers or in some other way) five more ones on to the seven; the result is then memorised, learnt by rote, for the procedure has no inner meaning. Similarly, we learn that 7 x 5 = 35 by counting on the fingers, etc., to one seven, adding on another, and repeating this five times, the result being likewise memorised. The trouble of this counting, of finding the sums or products, is eliminated by the completed addition and multiplication tables, which have only to be learnt by heart.

§ 452

Kant considers the proposition: 7 + 5 = 12 to be a synthetic proposition. 'We might indeed', he says, 'at first think (of course!) that it is a merely analytic proposition obtained by the law of contradiction from the concept of a sum of seven and five.' The concept of the sum means nothing more than the abstract determination that these two numbers are meant to be aggregated and, as numbers, in an external, that is, mechanical (begrifflose) fashion — that we are to count on from seven until the ones to be added on (their amount is fixed at five) have been exhausted; the result bears the otherwise familiar name of twelve. 'But', continues Kant, 'If we look more closely we find that the concept of the sum of 7 and 5 contains nothing save the union of the two numbers into one, and in all this there is no thinking at all about which is the single number which combines both' ... 'I may analyse my concept of such a possible sum as much as I please, still, I shall not find the twelve in it." With the thinking of the sum, with the analysis of the concept, the transition from the problem to the result has, of course, nothing whatever to do; and he adds, we must go outside these concepts and have recourse to intuition, to our five fingers and so on, thus adding on to the concept of seven the five units given in intuition. Five is, of course, given in intuition, that is, it is a wholly external aggregation of the arbitrarily repeated thought of one; but seven equally is not a concept; there are no concepts to go outside of or beyond. The sum of 5 and 7 means the mechanical [begrifflose] conjunction of the two numbers, and the counting from seven onwards thus mechanically continued until the five units are exhausted can be called a putting together, a synthesis, just like counting from one onwards; but it is a synthesis wholly analytical in nature, for the connection is quite artificial, there is nothing in it or put into it which is not quite externally given. The postulate that 5 be added to 7, bears the same relation to the postulate of simply counting, as does the postulate that a straight line be produced to the postulate that a straight line be drawn.

§ 453

Just as meaningless as the expression 'synthesis', is its characterisation as occurring a priori. Counting is not, of course, determined by sensation which, according to Kant's definition of intuition is all that remains over for the a posteriors, and counting is certainly an activity based on abstract intuiting, that is, an intuiting determined by the category of the one, and in which abstraction is made from all other determinations of sensation, no less than from concepts, too. The a priori is altogether too vague a characterisation; feeling, determined as impulse, sense, and so on, has in it the a priori moment, just as much as space and time, in the shape of spatial and temporal existence, is determined a posteriors.

§ 454

We may add in this connection that Kant's assertion of the synthetic nature of the foundations of pure geometry is equally without any solid basis. He admits that several are really analytic but only adduces one in support of his assertion that they are synthetic, namely, the axiom that the straight line is the shortest line between two points. 'For my notion of straightness contains nothing pertaining to magnitude, but only a quality; the notion of the shortest is, therefore, wholly an addition and cannot be inferred from any analysis of the notion of a straight line; we must therefore have recourse to intuition here which alone makes the synthesis possible.' But here again the question is not of a notion of straightness as such but of a straight line, and this is already something spatial and intuited. The determination (or, if you like, the concept) of the straight line is, after all, none other than that the line is absolutely simple, that is, in coming out of itself (the so-called movement of the point) the line is purely self-related, and its extension does not involve any alteration in its determination, or reference to another point or line outside itself; it is a simple direction purely internal to the line. This simplicity is indeed its quality, and should it seem difficult to define the straight line analytically this would be due solely to the simplicity and self-relation of the determination, and merely because reflection thinks of determining primarily in terms of a plurality, a determining through something else. But there is no inherent difficulty whatever in grasping this determination of the simplicity of extension within the line, of the absence of determination by anything else; Euclid's definition contains nothing else than this simplicity. But now the transition of this quality to the quantitative determination (of the shortest) which is supposed to constitute the synthetic element is wholly analytical. As spatial, the line is quantity in general; the simplest in terms of quantum is the least; and this predicated of a line is the shortest. Geometry can accept these determinations as a corollary to the definition; but Archimedes in his books on the sphere and cylinder' did the appropriate thing in making the said determination of a straight line into an axiom, in just as correct a sense as Euclid included the determination concerning parallel lines among the axioms, for the development of this determination into a definition would have required determinations not immediately spatial in character but of a more abstract qualitative kind, like simplicity, sameness of direction, and the like just mentioned. These ancients gave even to their sciences a plastic character, confining their exposition strictly to the peculiarity of their subject matter and therefore excluding what would have been heterogeneous to it.

§ 455

Kant's notion of synthetic a priori judgements — the notion of something differentiated which equally is inseparable, of an identity which is in its own self an inseparable difference, belongs to what is great and imperishable in his philosophy. Of course, this notion is also present in intuition since it is the Notion itself and everything is implicitly the Notion; but the determinations selected in those examples do not exhibit it. On the contrary, number and counting is an identity and the creating of an identity which is wholly and solely external, only a superficial synthesis, a unity of ones which are, in fact, posited as inherently not identical with one another but as external, each separate on its own account; the determination of the straight line as being the shortest between two points is based rather on the moment of a merely abstract identity possessing no difference within itself.

§ 456

I return from this digression to addition itself. The negative species of calculation corresponding to it, subtraction, is similarly the wholly analytical separation into numbers which, as in addition, are determined only as unequal generally relatively to one another.

§ 457

2. The next determination is the equality of the numbers which are to be counted. Through this equality they are a unit, and number thus acquires the difference of unit and amount. Multiplication is the task of counting up an amount of units each of which is itself an amount. Here it is immaterial which of the two numbers is called unit and which amount, whether we say four times three, where four is the amount and three is the unit, or conversely, three times four. We have already indicated above that the original finding of the product is effected by simple counting, that is, counting off on the fingers, etc.; the subsequent ability to state the product immediately rests on the collection of these products, on the multiplication table and on knowing it by heart.

§ 458

Division is the negative species of calculation with the same determination of the difference. Equally it is immaterial which of the two factors, the divisor or quotient, is taken as unit or amount. The divisor is taken as unit and the quotient as amount when the problem is to find out how often (amount) a number (unit) is contained in a given number; conversely, the divisor is taken as amount and the quotient as unit, when the problem is to divide a number into a given amount of equal parts and to find the magnitude of such part (of the unit).

§ 459

3. The two numbers which are determined as being related to each other as unit and amount are, as number, still immediate with respect to each other and therefore simply unequal. The further equality is that of unit and amount themselves; with this, the advance to equality of the determinations immanent in the determination of number is completed. On the basis of this complete equality, counting is the raising to a power (the negative version is extraction of the root) — and in the first instance the squaring of a number — the complete immanent determinedness of counting where (1) the several numbers to be added are the same, and (2) their plurality or amount is itself the same as the number which is posited a plurality of times, which is unit. There are no other determinations in the Notion of number which could give rise to a difference; nor can there be any further equalising of the difference immanent in number. Where a number is raised to a higher power than the square the continuation is formal: partly, with even exponents, there is only a repetition of squaring, and partly, with uneven powers, inequality again enters; to take the nearest example of the cube, although formally there is an equality of the new factor with both the amount and the unit, this factor is, as unit, unequal to the amount (the square, 3, against 3 times 3); this is still more evident in the cube Of 4 where the amount, 3, which indicates the number of times the unit is to be multiplied by itself, is different from the number itself standing for the unit. We have here in principle those determinations of amount and unit which, as the essential difference of the Notion, have to be equalised before number as a going-out-of-itself has completely returned into itself. The foregoing exposition also contains the reason why first, the solution of higher equations must consist on their reduction to quadratics, and secondly, why equations with odd exponents can only be formally determined and, just when the roots are rational they cannot be found otherwise than by an imaginary expression, that is, by the opposite of that which the roots are and express. From what has been said, it is clear that the arithmetical square alone contains an immanent absolute determinedness; for which reason equations with further formal powers must be reduced to it, just as in geometry the right-angled triangle contains an immanent, absolute determinedness which is expounded in Pythagoras' theorem, for which reason, too, all other geometrical figures must be reduced to it for their complete determination.

§ 460

A graded instruction based on a logically formed division of the subject treats of powers before it treats of proportions; it is true that the latter are connected with the difference of unit and amount which constitutes the determination of the second species of calculation, but they go beyond the one of the immediate quantum in which unit and amount are only moments; the further determination in the sphere of the immediate quantum still remains external to the quantum itself. In ratio, number is no longer an immediate quantum; it then possesses its determinateness as a mediation. The quantitative' relation will be considered in what follows.

§ 461

It cannot be said that the progressive determination of the species of calculation here given is a philosophy of them or that it exhibits, possibly, their inner significance; and for this reason, that it is not in fact an immanent development of the Notion. However, philosophy must be able to distinguish what is an intrinsically self-external material; the progressive determining of it by the Notion can then take place only in an external manner, and its moments, too, can be only in the form peculiar to their externality, as here, equality and inequality. It is an essential requirement when philosophising about real objects to distinguish those spheres to which a specific form of the Notion belongs, that is, spheres in which the Notion has an actual existence; otherwise the peculiar nature of a subject matter which is external and contingent will be distorted by Ideas, and similarly these Ideas will be distorted and made into something merely formal. But here, the externality in which the moments of the Notion appear in this external material, number, is the appropriate form; since these moments exhibit the subject matter in its fixed differences, and since, too, they contain no demand for speculative thinking and therefore appear easy, they deserve to be employed in text books of the elements.

Remark 2: The Employment of Numerical Distinctions for Expressing Philosophical Notions

§ 462

As we know, Pythagoras represented rational relationships (or philosophemata) by numbers; and more recently, too, numbers and forms of their relations, such as powers and so on, have been employed in philosophy for the purpose of regulating thoughts or expressing them. From an educational point of view, number has been regarded as the most suitable object of inner intuition and arithmetical operations with number have been held to be the mental activity in which mind brings to view its most characteristic relationships and in general, the fundamental relationships of essence. How far number can claim this high worth is evident from its Notion as now before us.

§ 463

We saw that number is the absolute determinateness of quantity, and its element is the difference which has become indifferent — an implicit determinateness which at the same time is posited as wholly external. Arithmetic is an analytical science because all the combinations and differences which occur in its subject matter are not intrinsic to it but are effected on it in a wholly external manner. It does not have a concrete subject matter possessing inner, intrinsic, relationships which, as at first concealed, as not given in our immediate acquaintance with them, have first to be elicited by the efforts of cognition. Not only does it not contain the Notion and therefore no problem for speculative thought, but it is the antithesis of the Notion. Because of the indifference of the factors in combination to the combination itself in which there is no necessity, thought is engaged here in an activity which is at the same time the extreme externalisation of itself, an activity in which it is forced to move in a realm of thoughtlessness and to combine elements which are incapable of any necessary relationships. The subject matter is the abstract thought of externality itself.

§ 464

As this thought of externality, number is at the same time the abstraction of the manifoldness of sense, of which it has retained nothing but the abstract determination of externality itself. In number, therefore, sense is brought closest to thought: number is the pure thought of thought's own externalisation.

§ 465

The mind which rises above the world of the senses and contemplates its own essence, when it seeks an element for its pure representation, for the expression of its essence, may therefore happen on number, this inner, abstract externality, before it grasps thought itself as this element and wins the purely spiritual expression for the representation of its essence. This is why we see number used for the expression of philosophemata early on in the history of philosophy. It forms the latest stage in that imperfection which contemplates the universal admixed with sense. The ancients were clearly aware that number stands midway between sense and thought. Aristotle quotes Plato as saying that the mathematical determinations of things stand apart from and midway between the world of the senses and the Ideas; they are distinguished from the former by being invisible (eternal) and unmoved, and from the latter by being a many and a like, where as the Idea is purely self-identical and within itself a one. A more detailed and profound reflection on this subject by Moderatus of Cadiz is quoted in Malchi Vita Pythagorae. That the Pythagoreans hit on numbers he ascribes to their inability to apprehend clearly in reason fundamental ideas and first principles, because these are hard to think and hard to express; numbers serve well as designations in instruction; in this among other things the Pythagoreans imitated the geometers who cannot express what is corporeal in thoughts and therefore use figures, saying 'this is a triangle', by which they do not mean that the visible drawing is to be taken for the triangle but only that it is a representation of the thought of it. Thus the Pythagoreans expressed the thought of unity, of self-sameness and equality and the ground of agreement, of connection and the sustaining of everything, of the self-identical, as a one. It is superfluous to remark that the Pythagoreans passed on from numbers to thought as a medium of expression, to the express categories of like and unlike, of limit and infinity; even in respect of these numerical expressions it is reported that the Pythagoreans distinguished between the Monas and the one ; the Monas they took as the thought, but the one as the number, and similarly, they took two for the arithmetical term and the Dyas (for this is what it seems to mean there) for the thought of the indeterminate. These ancients at the outset perceived quite correctly the inadequacy of number forms for thought determinations and equally correctly they further demanded in place of this substitute for thoughts the characteristic expression; how much more advanced they were in their thinking than those who nowadays consider it praiseworthy, indeed profound, to revert to the puerile incapacity which again puts in the place of thought determinations numbers themselves and number-forms like powers, the infinitely great, the infinitely small, one divided by infinity, and other such determinations, which are themselves often only a perverted mathematical formalism.

§ 466

In connection with the expression quoted above, that number stands between sense and thought, since, like the former, it is in its own self a many and an asunderness, it must be observed that the many itself is sense taken up into thought, is the category of what is in its own self external and so proper to sense. When richer, concrete veritable thoughts, when what is most alive and most active, what is comprehended only in its concrete relationships, when such are transposed into this element of pure self-externality, they become dead, inert determinations.

§ 467

The richer in determinateness and, therefore, in relationships thoughts become, the more confused and also the more arbitrary and meaningless becomes their representation in such forms as numbers. The one, the two, the three, the four, Henas or Monas, Dyas, Trias, Tetractys, have still some resemblance to the wholly simple abstract Notions; but when numbers are supposed to represent concrete relationships, it is vain to try to retain such resemblance. ®

§ 468

But thought is set its hardest task when the determinations for the movement of the Notion through which alone it is the Notion, are denoted by one, two, three, four; for it is then moving in its opposite element, one which is devoid of all relation; it is engaged on a labour of derangement. The difficulty in comprehending that, for example, one is three and three is one, stems from the fact that one is devoid of all relation and therefore does not in its own self exhibit the determination through which it passes into its opposite; on the contrary, one is essentially a sheer excluding and rejection of such a relation. Conversely, understanding makes use of this to combat speculative truth (as, for example, against the truth laid down in the doctrine called the trinity) and it counts the determinations of it which constitute one unity, in order to expose them as sheer absurdity — that is, understanding itself commits the same absurdity of making that which is pure relation into something devoid of all relation. When the trinity was so named it was not reckoned that understanding would consider the one and number to be the essential determinateness of its content. This name expresses contempt for the understanding, which has nevertheless confirmed itself in its conceit of clinging to the one and number as such, and has set it up against reason.

§ 469

To take numbers and geometrical figures (as the circle triangle etc., have often been taken), simply as symbols (the circle, for example, as a symbol of eternity, the triangle, of the trinity), is so far harmless enough; but, on the other hand, it is foolish to fancy that in this way more is expressed than can be grasped and expressed by thought. Whatever profound wisdom may be supposed to lie in such meagre symbols or in those richer products of fantasy in the mythology of peoples and in poetry generally, it is properly for thought alone to make explicit for consciousness the wisdom that lies only in them; and not only in symbols but in nature and in mind. In symbols the truth is dimmed and veiled by the sensuous element; only in the form of thought is it fully revealed to consciousness: the meaning is only the thought itself.

§ 470

But the perversity of employing mathematical categories for the determination of what belongs to the method or content of the science of philosophy is shown chiefly by the fact that, in so far as mathematical forms signify thoughts and distinctions based on the Notion, this their meaning has indeed first to be indicated, determined and justified in philosophy. In the concrete philosophical sciences philosophy must take the logical element from logic, not from mathematics; it can only be an expedient of philosophical incapacity which, instead of going to philosophy for the logical element, has recourse to the shapes assumed by the logical element in other sciences, many of which shapes are only adumbrations of that element, others even defective forms of it. Apart from this, the mere application of such borrowed forms is an external procedure; the application itself must be preceded by an awareness not only of their meaning but of their value, too, and such awareness can come only from reflecting on them, not from the authority of mathematics. But logic itself is such awareness and it strips these forms of their particularity which it renders superfluous and unnecessary; it is logic which rectifies these forms and alone procures for them their justification, meaning and value.

§ 471

As for the supposed primary importance of number and calculation in an educational regard, the truth of the matter is clearly evident from what has been said. Number is a non-sensuous object, and occupation with it and its combinations is a non-sensuous business; in it mind is held to communing with itself and to an inner abstract labour, a matter of great though one-sided importance. For, on the other hand, since the basis of number in only an external, thoughtless difference, such occupation is an unthinking, mechanical one. The effort consists mainly in holding fast what is devoid of the Notion and in combining it purely mechanically. The content is the empty one; the substantial content of moral and spiritual life in its various forms on which, as the noblest aliment, education should nurture the young mind, is to be supplanted by the blank one or unit; when such exercise is made the prime interest and occupation, the only possible outcome must be to dull the mind and to empty it of both form and substance. Calculation being so much an external and therefore mechanical business, it has been possible to construct machines which perform arithmetical operations with complete accuracy. A knowledge of just this one fact about the nature of calculation is sufficient for an appraisal of the idea of making calculation the principal means for educating the mind and stretching it on the rack in order to perfect it as a machine.

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